Existence and density conservation using a non-conservative approximation for Safronov-Dubovski aggregation equation

TL;DR

This paper investigates the existence and density conservation in the Safronov-Dubovski aggregation equation using a non-conservative approximation across various kernel coefficients, employing advanced mathematical tools.

Contribution

It introduces a non-conservative approximation approach to establish existence and density conservation for the equation with different kernel conditions.

Findings

Proved existence of solutions for multiple kernel types.

Identified conditions for mass conservation.

Applied advanced theorems like Helly's and De la Vallée-Poussin.

Abstract

The paper deals with the global existence and density conservation for the Safronov-Dubovski equation for three different coefficients $\phi$ such that $\phi_{i,j} \leq \frac{(i+j)}{\min\{i,j\}}$, $\phi_{i,j} \leq (i+j)$ and $\phi_{i,j} \leq (1+i+j)^{\alpha}$ $\forall$ $i,j \in \mathbb{N}$, $\alpha \in [0,1]$. The non-conservative approximation is applied to study the problem and results such as Helly's selection theorem and the refined version of the De la Vall\'ee-Poussin theorem are implemented to establish the existence of each case of the kernel. The article also focuses on the conditions for the conservation of mass per unit volume for such an equation.